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The tutorial below imports NumPy, Pandas, SciPy, and Random.
import plotly.plotly as py import plotly.graph_objs as go from plotly.tools import FigureFactory as FF import numpy as np import pandas as pd import scipy import random
A random walk can be thought of as a random process in which a tolken or a marker is randomly moved around some space, that is, a space with a metric used to compute distance. It is more commonly conceptualized in one dimension ($\mathbb$), two dimensions ($\mathbb^2$) or three dimensions ($\mathbb^3$) in Cartesian space, where $\mathbb$ represents the set of integers. In the visualizations below, we will be using scatter plots as well as a colorscale to denote the time sequence of the walk.
The jitter in the data points along the x and y axes are meant to illuminate where the points are being drawn and what the tendancy of the random walk is.
x = [0] for j in range(100): step_x = random.randint(0,1) if step_x == 1: x.append(x[j] + 1 + 0.05*np.random.normal()) else: x.append(x[j] - 1 + 0.05*np.random.normal()) y = [0.05*np.random.normal() for j in range(len(x))] trace1 = go.Scatter( x=x, y=y, mode='markers', name='Random Walk in 1D', marker=dict( color=[i for i in range(len(x))], size=7, colorscale=[[0, 'rgb(178,10,28)'], [0.50, 'rgb(245,160,105)'], [0.66, 'rgb(245,195,157)'], [1, 'rgb(220,220,220)']], showscale=True, ) ) layout = go.Layout( yaxis=dict( range=[-1, 1] ) ) data = [trace1] fig= go.Figure(data=data, layout=layout) py.iplot(fig, filename='random-walk-1d')
x = [0] y = [0] for j in range(1000): step_x = random.randint(0,1) if step_x == 1: x.append(x[j] + 1 + np.random.normal()) else: x.append(x[j] - 1 + np.random.normal()) step_y = random.randint(0,1) if step_y == 1: y.append(y[j] + 1 + np.random.normal()) else: y.append(y[j] - 1 + np.random.normal()) trace1 = go.Scatter( x=x, y=y, mode='markers', name='Random Walk', marker=dict( color=[i for i in range(len(x))], size=8, colorscale='Greens', showscale=True ) ) data = [trace1] py.iplot(data, filename='random-walk-2d')
We can formally think of a 1D random walk as a point jumping along the integer number line. Let $Z_i$ be a random variable that takes on the values +1 and -1. Let this random variable represent the steps we take in the random walk in 1D (where +1 means right and -1 means left). Also, as with the above visualizations, let us assume that the probability of moving left and right is just $\frac$. Then, consider the sum
where S_n represents the point that the random walk ends up on after n steps have been taken.
To find the expected value of $S_n$, we can compute it directly. Since each $Z_i$ is independent, we have
but since $Z_i$ takes on the values +1 and -1 then
$$ \begin \mathbb(Z_i) = 1 \cdot P(Z_i=1) + -1 \cdot P(Z_i=-1) = \frac - \frac = 0 \end $$
Therefore, we expect our random walk to hover around $0$ regardless of how many steps we take in our walk.
